# Trigonometric Tapestry

> Six nested trigonometric functions, each turned by a sixth root of unity, summed and painted over the complex plane.

Nesting one trigonometric function inside another, as in $\sin(\cos z)$, gives a function of a complex variable with a surprisingly rich structure: zeros and poles arranged in rows, and essential singularities where an inner function has a pole. Adding several such functions together mixes their patterns into a single field.

Here $f(z) = \sum_{n=0}^{5} e^{in\pi/3}\, g_n(z)$, where $g_n$ runs through $\sin(\cos z)$, $\cos(\sin z)$, $\tan(\cot z)$, $\cot(\tan z)$, $\sec(\csc z)$ and $\csc(\sec z)$, and each term is turned by a sixth root of unity. Every pixel $z = x + yi$ is colored by $(1 - M)\,z + M f(z)$: hue shows the argument and lightness the modulus, so zeros and poles appear as points where all the colors meet. As $M$ animates from $0$ to $1$, the plain color wheel of the identity map morphs into the full sum, which repeats with period $2\pi$ along the real axis. The string of tiny features on the real axis comes from $\tan(\cot z)$ and its neighbors, which have an essential singularity at every multiple of $\pi$.

After a Desmos graph by DragonFire28.

- [Open the interactive version](https://graph-paper.io/showcase/en/complex-trig-sum)
- [Learn more on Wikipedia](https://en.wikipedia.org/wiki/Domain_coloring)